m36

Prompt

RESEARCH QUESTION (plane continua / composants and accessibility of an indecomposable boundary under a homeomorphism; SS-COF (no failing door at cofinal depth of a canonical KP strip) in composant form: by Theorem IMP* it says that the ends of deep doors lie in ONE composant; which law of the action of h and of its perfect extension on composants and accessible composants near the outchannel forces this?). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference; unchecked?), LAB (from this prompt, status as given), CONJECTURE, OPEN, or RESTATEMENT (a gap reworded). If a definition is ambiguous, state your reading in one line and continue. Say where your argument stops. A. DEFINITIONS Setting (compressed from the lab lanes b_opus_imptransport, b_opus_ssdyn, codex_sscof and microeval m24; UNAUDITED unless said). - X is a nondegenerate nonseparating plane continuum, indecomposable (BFMOT Thm 6.1.4 under the standing hypotheses, SOURCE-lab), so X has empty interior and U := C \ X is a simply connected domain through infinity. h: X -> X is a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K has h^2(K) inside K) and m_1 := min_x d(x, h x) > 0 (so h has no fixed point and no point of period 2). F, G are perfect plane extensions of h, h^-1 (continuous, proper, F|X = h; G is not F^-1 off X; F(s) may meet X). Each has a unique outchannel E_F, E_G (prime ends of U with principal set X), E_F != E_G; Theta_X := {prime ends with impression X} (closed, contains E_F, E_G). - Composants. a ~ b iff some proper subcontinuum of X contains both; the classes are the composants, each dense, each a countable union of proper subcontinua (hence meagre); C(x) := the composant of x. h permutes composants: h C(x) = C(h x). A point is accessible if it is the end of a crosscut; Acc(X) is the set of accessible points and AC the set of composants meeting Acc(X) ("accessible composants"). - A crosscut s is an open arc in U whose closure is an arc with distinct ends a, b in X; Sh(s) is the bounded component of C \ (X u s); K_s := the smallest subcontinuum containing a and b, so K_s = X iff a !~ b. A door s is M_F (M_G) if Sh(s) misses a tail of the external ray of E_F (E_G); a door FAILS if it is M_F, M_G and a !~ b. A canonical KP strip near E_F is a nested chain of Kuratowski-Pelczynski chords defining E_F, depth n = the n-th chord. SS-COF: some n_0 has no failing door of depth >= n_0. FT: every crosscut q with diam q <= m_1/8 that is M_F and M_G has a !~ b false. B. LEMMAS (status; where used) Known and derived here (UNAUDITED; the three DERIVED-here items are one-line facts to be checked, not imported). - Theorem IMP* (b_opus_imptransport, DERIVED; confirmed independently by m24: Argon 2.3, GPT-6 Sol): for ANY crosscut c with a !~ b the closed cap of c contains a prime end with impression X. Hence SS-COF <=> no Theta_X point in the caps of deep doors <=> (indecomposability) the two ends of every deep door lie in one composant. This is the form of the target; it is a RESTATEMENT, not an input. - Composant cocycle (m24, GPT-6 Sol, PROVED): for a composant C and an oriented crosscut a -> b, J_C := 1_C(a) - 1_C(b); additive along successive pairs; nonzero for the failing doors. It sees K_s = X but is blind to dynamics and proves nothing alone. - Composant transport (b_opus_imptransport 4.2, DERIVED): a !~ b iff h a !~ h b; so the image pair (h a, h b) of a failing door is composant separated; by IMP* ANY join of the image pair carries a Theta_X point in its cap. Prop EXC: if F(s) meets X, either (E1) some excursion of F(s) joins two composants (then IMP* applies to it) or (E2) every excursion stays in one composant and the class changes only at accumulation points of F(s) cap X (a Knaster-type Cantor transition); plane topology does not exclude (E2); the lemma E1-LAW (at a failing M_F door F(s) has a composant-changing excursion) is OPEN. - HOM circle model (b_opus_imptransport 6.3, DERIVED by hand, Tier 0): on a prime-end circle R/Z with E_F = 0, E_G = 1/2, a circle homeomorphism phi with phi(I_n) = I_{n-1} on door caps I_n = (4^-n, 1.5 * 4^-n), composant classes of the door ends shifted n -> n-1 by phi, has a failing door at every depth and satisfies every impression/composant transport law. So an equivariant composant law cannot prove SS-COF; any proof must use an input that this model violates (a fixed-point-free h, which no plane homeomorphism with a nonseparating invariant continuum can be, by Cartwright-Littlewood). - DERIVED-here (i): x ~ h^k x iff h^k C(x) = C(x). (ii) If h^2 x ~ x and L is a proper subcontinuum containing x and h^2 x, then L_j := L u h^2 L u ... u h^(2j) L is connected and proper (each step meets the last in h^(2j+2) x; a union of two intersecting proper subcontinua of an indecomposable continuum is proper), all inside C(x), and cl(union_j h^(2j) L) is a continuum K with h^2 K inside K, so K = X by cm: the forward h^2-orbit of L is dense. (iii) ~ is F_sigma in X x X (a countable union of closed relations "same component of X minus a ball B(y, 1/k)"), so separated pairs form a G_delta. - Accessibility under h (lane 974, H-NH7, OPEN-OBSTRUCTION, UNAUDITED): variation of small crosscuts never sees whether image points are accessible (R-VAR); on the class FCLS h(Acc) in Acc implies h(Acc) = Acc, and any accessibility change moves a whole accessible arc component onto an inaccessible one (a CAD hole). - Accessible composants vary (topology alone allows both extremes). Bucket handle (RECALL, not checked on disk): in the standard planar embedding all accessible points lie in one composant, so no crosscut has full trace. Lewis's continuum P_s (OV11 Ex. 5.4, SOURCE as quoted in the lab audit loop_audit_web_r28_thin_build_sonnet line 25, no h): no proper subcontinuum contains two accessible points, so ANY two accessible points lie in distinct composants and EVERY crosscut has full trace. C. THE TARGET SS-COF in composant form: in every R-instance, the ends of every door of depth >= n_0 of a canonical KP strip near E_F lie in one composant (equivalently, AC restricted to the deep door ends is one composant). D. ASK Part 1 (law). The failing doors give accessible points in distinct composants, so |AC| >= 2 is necessary for a failing door. Decide which of the following laws of an R-instance can bear on this, state it precisely and say what it uses: (L1) invariance: is AC invariant under h (accessibility is not preserved by the perfect F; say exactly what F(s) cap X does to it, using (E1)/(E2)); (L2) orbit structure: by DERIVED-here (i)-(ii) a composant with h^2 C = C is h^2-invariant with dense forward orbits of its subcontinua, so what do cm, m_1 > 0 and E_F != E_G say about invariant, periodic and wandering composants, and about the h-orbit of the composants of the ends of the doors s_n (all failing doors at once, bi-infinitely); (L3) a Baire argument on X x X with the G_delta set of separated pairs and the meagre composants; (L4) a law near the outchannel: how the composant of a section end or door end at depth n is related to the composant of its h-image at a shallower depth, given the outchannel pattern (a section g has g in Sh F(g), var(F, g) = -1). Name the one you use and why the HOM circle model (composant classes shifted by phi) does not already satisfy it. Part 2 (carry it out). Prove SS-COF in composant form, or reduce it to ONE named lemma with gate G1-G4, or show that the law cannot see cofinal composant split (give the abstract data). Say where the argument uses cm of h^2 and where it uses m_1 > 0, and whether it needs F homeomorphic. An argument that uses only the h-orbit of the pair (a_n, b_n) and the equivalence ~ is the HOM circle model again and must be excluded. Part 3 (builder or decisive test). Build abstract data for cofinal failing doors: a set Comp with an h-permutation pi, a subset AC, sequences C_n^-, C_n^+ in AC of composants of the left/right ends of the depth-n door with C_n^- != C_n^+, E_F != E_G, and check every law of Part 1, cm-type (no finite pi-cycle structure contradicting DERIVED-here (ii)), m_1 > 0 and the finite-depth consistency. Record the first law your data violates, or report consistency with the exact list checked. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT; UNKNOWN if unsure a result exists. Take your time and be extremely meticulous: check every step, every hypothesis and every quantifier before you write it down. This is a hard open problem, and I believe in you. E. CONTROLS (all must hold) (C1) Two h-free extremes: the standard bucket handle (one accessible composant, no full-trace crosscut) and Lewis's P_s (all accessible points in distinct composants, every crosscut full trace). The law you use must say at which hypothesis of an R-instance (h, m_1, cm, perfect F) it separates an R-instance from P_s, and must not derive "one accessible composant" from topology alone (P_s refutes that). (C2) The HOM circle model: composant classes shifted by phi, failing door at every depth; your law must be violated by it or you have not used anything new. (C3) Finite depth: a threshold depending on n does not prove SS-COF. (C4) A host where h has fixed points (planar odometer-end host Y_2): the law must not conclude there, and you must say which hypothesis fails. F. DEAD ENDS (not progress) (1) Variation, index and degree counts at deep doors; KP enclosing curves (codex_sscof: all zero). (2) cm on Hausdorff limits of pocket traces: the trace is X (m24, all five answers). (3) Impression transport, TT, JOIN, LIM-FT: satisfied by the HOM circle model (b_opus_imptransport 6.3). (4) IMP3 or FT as input: circular (IMP3 <=> FT <=> SS strip-free). (5) Invariants through H_1 of U minus (R_F u R_G): blind (Argon, m24). (6) Composant cocycle J_C alone: tautological detector (m24). (7) A Cartwright-Littlewood index of F on one curve: assigned to another lane. RESTATEMENT GATE (mandatory; answered BEFORE the headline, in its own section). If your verdict reduces the node N to a residual lemma L (with or without other proved pieces), you must give ALL of: (G1) the exact statement of L and of N side by side, with the same standing hypotheses; (G2) a proof that L implies N (the reduction); (G3) a reason L is STRICTLY weaker than N: either an explicit configuration (or consistent abstract data meeting every standing law) where L holds and the conclusion of N is not yet available, or the named hypothesis of L that the negation of N does NOT supply; (G4) the line of your argument where L's own hypothesis (the defect, the witness, the scale) is actually used; if it is never used, L is a restatement. If G3 or G4 fails, the headline word is RESTATEMENT, not REDUCED, and the answer reports the proved building blocks only. A headline REDUCED without G1-G4 counts as a failed answer. G. FORMAT Line 1 exactly: VERDICT: SS-COF PROVED (a named composant/accessibility law of an R-instance forces deep door ends into one composant; proof) | COFINAL COMPOSANT SPLIT CONSISTENT (explicit composant data with cofinal doors whose ends lie in distinct composants meeting every listed law; first missing global law named) | REDUCED (to one named lemma, gate G1-G4 passed) | PARTIAL (one named sub-claim settled) | OPEN. Then SUMMARY (<= 100 words), the argument (<= 2,000 words), STILL OPEN EXACTLY. Short of effort: line 1 + summary; an honest OPEN is valid. No finite computation as proof; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: restating SS-COF as "door ends lie in one composant" and calling that a law (it is IMP* applied to the target: RESTATEMENT); using a variation or index count at deep doors; using cm on pocket traces (the trace is X); using impression transport, TT or JOIN (the HOM circle model satisfies them); assuming F is a plane homeomorphism, or that F preserves accessibility (F(s) may meet X); treating the standard Knaster embedding as an R-instance; treating a finite depth as a cofinal configuration; claiming DP-K, L2 or PER